Invex function

In <a href="/facts/Vector_calculus/mXKIJLJE">vector calculus</a>, an invex function is a <a href="/facts/Differentiable_function/jQqTgLk1">differentiable function</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
 from 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 to 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 for which there exists a vector valued function 
 
 
 
 η
 
 
 {\displaystyle \eta }
 
 such that

f
        (
        x
        )
        −
        f
        (
        u
        )
        ≥
        η
        (
        x
        ,
        u
        )
        ⋅
        ∇
        f
        (
        u
        )
        ,
        
      
    
    {\displaystyle f(x)-f(u)\geq \eta (x,u)\cdot \nabla f(u),\,}

for all x and u.
Invex functions were introduced by Hanson as a generalization of <a href="/facts/Convex_function/IbzG5SLF">convex functions</a>. Ben-Israel and Mond provided a simple proof that a function is invex if and only if every <a href="/facts/Stationary_point/f56GSgBj">stationary point</a> is a <a href="/facts/Global_minimum/ADlgnyzV">global minimum</a>, a theorem first stated by Craven and Glover.
Hanson also showed that if the objective and the constraints of an <a href="/facts/Optimization_problem/QzwJfFEE">optimization problem</a> are invex with respect to the same function 
 
 
 
 η
 (
 x
 ,
 u
 )
 
 
 {\displaystyle \eta (x,u)}
 
, then the <a href="/facts/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions/ly7g2zST">Karush–Kuhn–Tucker conditions</a> are sufficient for a global minimum.

Invex function open-in-new

Invex function